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Symmetric Dyck Paths and Hooley’s \(\varDelta \)-Function

  • José Manuel Rodríguez CaballeroEmail author
Conference paper
  • 389 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)

Abstract

Hooley [6] introduced the function
$$ \varDelta (n) := \max _{u \in \mathbb {R}}\# \left\{ d | n: \quad u < \log d \leqslant u+1 \right\} , $$
where \(\log \) is the natural logarithm. Changing the base of the logarithm from e to an arbitrary real number \(\lambda > 1\), we define
$$ \varDelta _{\lambda }(n) := \max _{u \in \mathbb {R}}\# \left\{ d | n:\quad u < \log _{\lambda } d \leqslant u+1 \right\} . $$
The aim of this paper is to express \(\varDelta _{\lambda }(n)\) as the height of a symmetric Dyck path defined in terms of the distribution of the divisors of n.

Keywords

Dyck path Palindrome Hooley’s \(\varDelta \)-function 

Notes

Acknowledgement

The author thanks S. Brlek, C. Kassel and C. Reutenauer for they valuable comments and suggestions concerning this research. Also, the author want to express his gratitude to H. F. W. Höft for the useful exchanges of information.

References

  1. 1.
    Blondin-Massé, A., Brlek, S., Garon, A., Labbé, S.: Combinatorial properties of \(f\)-palindromes in the Thue-Morse sequence. Pure Math. Appl. 19(2–3), 39–52 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Erdös, P., Nicolas, J.L.: Méthodes probabilistes et combinatoires en théorie des nombres. Bull. Sci. Math. 2, 301–320 (1976)zbMATHGoogle Scholar
  3. 3.
    Fine, N.J.: Basic Hypergeometric Series and Applications, vol. 27. American Mathematical Soc., Providence (1988)zbMATHGoogle Scholar
  4. 4.
    Hall, R.R., Tenenbaum, G.: Divisors. Cambridge Tracts in Mathematics, vol. 90. Cambridge University Press, Cambridge (1988).Google Scholar
  5. 5.
    Höft, H.F.W.: On the symmetric spectrum of odd divisors of a number. https://oeis.org/A241561/a241561.pdf
  6. 6.
    Hooley, C.: On a new technique and its applications to the theory of numbers. Proc. London Math. Soc. 3(1), 115–151 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kassel, C., Reutenauer, C.: Counting the ideals of given codimension of the algebra of Laurent polynomials in two variables. arXiv preprint arXiv:1505.07229 (2015)
  8. 8.
    Kassel, C., Reutenauer, C.: Complete determination of the zeta function of the Hilbert scheme of \(n\) points on a two-dimensional torus. arXiv preprint arXiv:1610.07793 (2016)
  9. 9.
    Rodríguez Caballero, J.M.: On a function introduced by Erdös and Nicolas (To appear)Google Scholar
  10. 10.
    Sloane, N.J.A., et al.: The on-line encyclopedia of integer sequences (2012)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Université du Québec à MontréalMontréalCanada

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